Tailed Minimum-Variance Hedge Ratios for Futures
Summary
The document examines Hull’s two formulations for choosing a minimum-variance futures hedge. One uses the correlation and standard deviations of absolute spot and futures price changes, then scales the hedge ratio by the quantities of the asset and futures contract. The alternative uses percentage price changes and scales by the values being hedged and represented by each contract. The question asks whether this percentage-based formulation is inherently superior, whether it applies to forwards, and how it relates to daily futures settlement.
It also distinguishes a further adjustment: discounting the hedge count to reflect the timing difference between immediate futures settlement and spot exposure realized later. The document raises, but does not answer, whether this interest-rate adjustment is distinct from the percentage-change formulation or the central meaning of tailing a hedge. Thus it presents useful conceptual distinctions and formulas, but offers no resolution, derivation, empirical comparison, or guidance on when to choose each method.
Key ideas
- The untailed minimum-variance hedge ratio uses absolute price changes and contract quantities.
- A percentage-change formulation scales the hedge using asset and contract values.
- The question is whether the percentage-based formulation is tied specifically to daily futures settlement.
- A separate adjustment discounts the hedge count for the timing of futures and spot cash flows.
- The document poses these distinctions without providing a definitive answer.
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Full text
# Tailing the Hedge for Minimum Variance Hedge Ratio (Hull, 10ed)
# Tailing the Hedge for Minimum Variance Hedge Ratio (Hull, 10ed)
I am an amateur reading Hull's Options, Futures and other Derivatives.
I have encountered an issue similar to the one here: How to tail a hedge? (Question 3.26 from Hull, edition 10).
The author initially presents a formula for the minimum variance hedge ratio for future contracts:
$$ h^* = \rho\frac{\sigma_S}{\sigma_F} $$ and the corresponding optimum number of contracts is: $$N^*=h^*\frac{Q_A}{Q_F}$$
where the quantities involved ($\rho,\sigma_S,\sigma_F$) are taken with respect to absolute changes in spot/future prices.
He then claims that to take into account daily settlement of futures, an alternative formula is to be used, necessary for "tailing the hedge":
$$ \hat{h} = \hat{\rho}\frac{\hat{\sigma_S}}{\hat{\sigma_F}} $$ and correspondingly: $$\hat N= \hat{h}\frac{V_A}{V_F}$$
Here, the difference is that our values with hats are taken with respect to percentage changes in spot/future prices, and we now deal with the ratio between values instead of quantities to find number of contracts.
He mentions that this is the first part of "tailing the hedge". In the second part, we divide $\hat{N}$ by $(1 + \text{interest rate})$ to take advantage of the fact that future payoffs are immediate, but the payment/payoffs from changes in spot prices are realised at some fixed point in the future
I've attempted to derive these independently, to some success. However, some points confuse me:
To the best I can fathom, the second set of equations is not necessarily unique to daily settlement, but rather an entirely different method of calculating the MVHR, looking towards the correlation between percentage changes in spot/futures prices, as opposed to seeking correlation between absolute changes in the first method.
- Is this not simply a "better" method that seeks to account for changes as a percentage of the initial value of the asset? Online resources present either one of the methods, seemingly arbitrarily with no explanation.
- Is this method not also applicable to futures contracts without daily settlement? (i.e. in forward contracts)
- Are the the two parts of "tailing the hedge" being wrongly conflated? I believe that the first part allows for adjustments to the optimal hedge ratio daily with varying spot/future prices, something not offered by the first formula (where quantities of our asset do not vary with time). However, the second part (interest rate) by itself appears to be the more widely accepted notion of "tailing the hedge", and is applicable irrespective of which of the two formulae is used.
This is my first time posting a question, and I apologize if my question is unclear or too basic.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.